The rigged eye does not close completely
It's a weights problem.
This tutorial is quite advanced and relates to a dense mesh properly and evenly modeled because it's automatically weighted.
As I can see from your 2nd picture (in example) even the closer vertices of the lower eyelid don't follow exactly the closer bones: it means that those vertices are weighted also in other Bones/vertex groups. Probably your mesh needs some manual weighting to work properly: if a vertex is weighted (any value of weight) to a bone/vertex group only, it follows EXACTLY the bone movement.
The button assign gives to the selected vertices the weight indicated below (from 0.0 to 1.0). The button remove removes the selected vertices from the vertex group. The button select adds to the current selection all the vertices of that vertex group. The button deseletc removes from the current selection all the vertices of the vertex group (but those deselected vertices are still assigned and weighted to the current vertex group, so - if you click select - they appear again). In edit mode, press N for the properties tab, select a vertex and you can find the "vertex weight" tab that shows you all weights and all  vertex groups your vertex is in (sorry for my english, I'm from Italy).